Method for accurate control of transmit pattern of a digital-analog hybrid array

Abstract

The application discloses a transmit pattern accurate control method for a digital-analog hybrid array. A target function of maximizing a ratio of a minimum main lobe level to a peak side lobe level is researched, and under main lobe ripple constraint, analog weight constant modulus constraint and digital weight power constraint, analog weight and digital weight are jointly designed by using an alternating direction penalty method (ADPM). In the ADPM framework, a backtracking projection gradient descent method is adopted to solve an analog weight sub-problem with constant modulus constraint; and a Lagrange multiplier method is adopted to solve digital weight. Numerical results show that the method can produce better main lobe ripple and lower peak side lobe level, and has good beam pattern accurate control performance.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing technology, specifically, it is a method for precise control of the transmission pattern of a hybrid analog digital (HAD) array. Background Technology

[0002] In modern radar and millimeter-wave communication, the application of large-scale transmit antenna arrays is crucial, serving to enhance target reception energy and compensate for the high path loss of millimeter-wave signals. However, fully digital (FD) architectures, with each antenna equipped with an independent radio frequency (RF) link, suffer from excessively high hardware costs and significantly increased complexity. Hybrid analog-digital (HAD) architectures have emerged as a promising alternative, using phase shifters to achieve flexible connections between RF links and antenna elements, offering near-FD array performance at a relatively low hardware cost. It operates by using a small number of RF links for digital beamforming, while simultaneously connecting antenna elements and RF links using numerous low-cost analog phase shifters to complete analog beamforming. Hybrid analog-digital (HAD) arrays have become a promising approach for realizing large-scale transmit arrays due to their trade-off between system performance and hardware complexity.

[0003] However, precise control of the emission pattern of HAD arrays faces a series of challenges. Because all analog weights in the HAD array are subject to constant modulus constraints, and analog and digital weights are heavily coupled in the objective function, emission pattern synthesis becomes a difficult problem to solve. Existing optimization algorithms often have limitations in handling this problem, such as the inability to effectively handle high-dimensional variables and the difficulty in balancing the relationship between local and global optima. Therefore, to achieve precise control of the emission pattern of HAD arrays, a novel algorithm capable of effectively solving high-dimensional, non-convex optimization problems needs to be developed. Summary of the Invention

[0004] The purpose of this invention is to propose a method for precise control of the transmission pattern of HAD arrays.

[0005] The technical solution to achieve the objective of this invention is: a method for precise control of the transmission pattern of an HAD array, comprising the following steps:

[0006] Step 1: For the precise control of the transmit pattern of an HAD array, taking the maximization of the ratio of the minimum main lobe level to the peak sidelobe level as the criterion, the non-convex optimization problem model for precise control of the transmit pattern is as follows:

[0007]

[0008] in, The discrete angles represent the side lobe regions. This is the side lobe region. Discrete angles of the main lobe region Main lobe region For angle HAD array steering vector at the location, The HAD array is a mixed weight vector. For the digital weights of the HAD array, For the simulated weights of the HAD array, The number of digital subarrays. The number of elements in each simulated subarray. express A dimensional vector of all 1s;

[0009] Step 2: Control the main lobe ripple within the set range, and introduce auxiliary variables to further express the non-convex optimization problem model of Step 1 as follows:

[0010]

[0011] in, Indicates the minimum main lobe level. Indicates the peak sidelobe level. Indicates the main lobe ripple. and These represent the number of discrete angle points in the main lobe region and the side lobe region, respectively.

[0012] Step 3: Introduce auxiliary variables and ,definition , Constructing the augmented Lagrange function

[0013]

[0014] in, , and These are introduced auxiliary variables. and It is a dual variable. and Let the penalty factor be ; then the optimization problem model after step 2 is rewritten as:

[0015]

[0016] Step 4: Based on the Alternating Direction Penalty Method (ADPM) framework, the optimization problem model in Step 3 is decomposed into several easily solvable subproblems, specifically:

[0017]

[0018]

[0019]

[0020]

[0021]

[0022]

[0023]

[0024] In this context, all variables with a superscript of k+1 represent the results of the k+1th iteration, and all variables with a superscript of k represent the results of the kth iteration. , , and Let be the penalty factor transformation coefficient, and satisfy . , , This represents the maximum value of the penalty factor;

[0025] Step 5: Solve the subproblems in Step 4, including simulating the weights. The solution is obtained iteratively using the backtracking projection gradient method, with digital weights. The closed-form solution is given using the Lagrange multiplier method, with auxiliary variables... and Closed-form solutions are given for all cases; finally, the optimal digital weights and analog weights are obtained to achieve the problem of precise control of the emission pattern of the HAD array.

[0026] Compared with existing technologies, the significant advantages of this invention are as follows: This invention reduces the sidelobe level and obtains a high-gain, stable main lobe by optimizing the power distribution of the main lobe and sidelobes; the invention reduces computational complexity and improves the feasibility of practical applications through optimized algorithm design. Due to the superior performance of the ADPM framework, the proposed algorithm also exhibits superior performance in precise control of the transmission pattern compared to other representative methods. Attached Figure Description

[0027] Figure 1 This is a flowchart of the precise control method for the emission pattern of HAD arrays proposed in this invention.

[0028] Figure 2This is the beam pattern of a single main lobe flat-top beam.

[0029] Figure 3 This is the beam pattern of a double main lobe flat-top beam. Detailed Implementation

[0030] The present invention, namely a method for precise control of the transmission pattern of an HAD array, is further described below with reference to the accompanying drawings and embodiments.

[0031] This invention relates to a precise transmit beamforming control method for HAD arrays. The method proposes a precise transmit beamforming algorithm based on ADPM (Advanced Dynamic Beamforming Process). Under constant-mode analog weights, main lobe ripple constraints, and digital weight power constraints, the algorithm synthesizes the desired beam pattern by maximizing the minimum main lobe level and the peak sidelobe level. The algorithm first introduces auxiliary variables to simplify fractional constraints, and then, within the ADPM framework, introduces auxiliary variables to alternately optimize the analog and digital weights. In each optimization, the backtracking projection gradient descent method is used to solve the subproblem with constant-mode constraints. The specific implementation steps are as follows:

[0032] Step 1: Consider the far-field narrowband signal of a HAD uniform linear array with a partially connected structure, where each analog subarray is composed of... It consists of several isotropic antenna elements, with the spacing between each element being [missing information]. Each analog subarray is connected to one RF link. Therefore, the total steering vector of the HAD array is expressed as:

[0033]

[0034] in, Indicates the angle of each simulated subarray. The guide vector at the location; The steering vector representing the digital link. Indicates the carrier wavelength.

[0035] Define the weight of the nth element on the mth simulated subarray as: , , The simulated weights of the HAD array are then expressed as:

[0036]

[0037] Define the RF link weight corresponding to the m-th analog subarray as: The digital weights of the HAD array are then expressed as:

[0038]

[0039] Therefore, the hybrid weights of the HAD array are equivalently represented as:

[0040]

[0041] The HAD array transmission pattern is

[0042]

[0043] Since the simulated weights in the HAD array need to maximize transmission efficiency, the following constraints must be satisfied:

[0044]

[0045] The digital weights of the HAD array must satisfy the total power constraint:

[0046]

[0047] To concentrate the transmitted signal energy more in the main lobe, this invention establishes a precise control model for the transmission pattern of the HAD array based on maximizing the ratio of the minimum main lobe level to the peak sidelobe level. The specific non-convex optimization model is as follows:

[0048]

[0049] In the formula, For the angle of the side lobe region, This is the side lobe region. From the angle of the main lobe region, Main lobe region For angle The guide vector at that location, For a mixed weight vector, For numerical weights, To simulate weights, The number of digital subarrays. The number of array elements for each simulated subarray.

[0050] Step 2: To enhance the stability of target detection performance in the main lobe region of the transmitted beam, the main lobe ripple needs to be controlled within a set range. Represent the main lobe ripple and define... Minimum main lobe level, This represents the peak sidelobe level. Therefore, the optimization problem model in step 1 can be further expressed as:

[0051]

[0052] In the formula, Indicates the minimum main lobe level. This represents the peak sidelobe level.

[0053] Step 3: Introduce auxiliary variables and ,definition , Constructing the augmented Lagrange function :

[0054]

[0055] in, , and It is a dual variable. and Let be the penalty factor. Then the optimization problem model in step 2 can be equivalently rewritten as:

[0056]

[0057] Step 4: Based on the ADPM framework, the optimization problem model in Step 3 is decomposed into several easily solvable subproblems, specifically as follows:

[0058]

[0059]

[0060]

[0061]

[0062]

[0063]

[0064]

[0065] In this context, all variables with a superscript of k+1 represent the results of the k+1th iteration, and all variables with a superscript of k represent the results of the kth iteration. , , and Let be the penalty factor transformation coefficient, and satisfy . , , This represents the maximum value of the penalty factor.

[0066] Step 5: Solve each sub-problem in Step 4 to obtain the optimal simulated weights and digital weights. The specific process is as follows:

[0067] Step 5.1: Ignore the constant term and... Equivalent representation is In the formula, all variables with superscript k represent the k-th iteration, and ,

[0068] ,

[0069] ;

[0070] The transformed subproblem is solved using the backtracking projection gradient descent method to obtain the simulated weights for the (k+1)th iteration. The specific solution process is as follows:

[0071] Step 5.1.1: Obtain the objective function for the j-th iteration. The gradient is Its negative gradient direction is ;

[0072] Step 5.1.2: Adaptively update the step size using backtracking search. The specific method is as follows:

[0073] Initialize step size Using the formula Iteration step size, This is the step size coefficient. When the backtracking condition is met, the iteration stops, and the current iteration position is set to... The final step size for the j-th round is defined by the backtracking condition as follows:

[0074]

[0075] Indicates the vector Each element is projected onto the unit circle, i.e. ;

[0076] Step 5.1.3: Solve for the simulated weights in the (j+1)th iteration using the following formula:

[0077]

[0078] Determine whether the iteration termination condition is met, i.e., whether it is satisfied. , This is the algorithm's termination tolerance; if it is met, the loop stops, and the current value is set to zero. Use the optimal simulation weight for the (k+1)th round; otherwise, return to step 5.1.1.

[0079] Step 5.2: Solve To obtain the numerical weights for the (k+1)th iteration, the specific steps are as follows:

[0080] Ignoring the constant term, the subproblem can be equivalently represented as

[0081]

[0082] in, , , Represents an N x 1 column vector. Indicates the vector Diagonalize to a matrix. Define the Lagrange function as:

[0083]

[0084] in, These are Lagrange multipliers. According to the first-order optimality condition of the Lagrange function, we can obtain...

[0085]

[0086] definition The matrix decomposition expression is as follows ,in, This represents a unitary matrix composed of eigenvectors. Let represent a diagonal matrix composed of eigenvalues, and Redefining and order Then the equation can be simplified to

[0087]

[0088] Solving for the optimal Lagrange multipliers using the line search method Substitute the Lagrange multipliers into the following formula:

[0089]

[0090] Obtain the numerical weights for the (k+1)th iteration. .

[0091] Step 5.3: Solve Obtain the auxiliary variables for the (k+1)th iteration. and peak sidelobe level The specific process is as follows:

[0092] Ignore irrelevant terms and let The subproblem is transformed into:

[0093]

[0094] First fix The following formula can be used to solve for it. :

[0095]

[0096] when hour, ,otherwise ; Define respectively for Given an ascending set and removing duplicates; let , , Introduction , The objective function in the subproblem is transformed into a piecewise function:

[0097]

[0098] in It can be represented as:

[0099]

[0100] The first derivative is

[0101]

[0102] make , , ,but The two roots are

[0103]

[0104] Then piecewise function In the interval The minimum value can be obtained by the following formula.

[0105]

[0106] The optimal peak sidelobe level for the (k+1)th iteration is obtained by selecting the global minimum value from all L+1 local piecewise functions.

[0107]

[0108] Will Substitute and solve In the formula, we obtain the auxiliary variable for the (k+1)th iteration. .

[0109] Step 5.4: Solve Obtain the auxiliary variables for the (k+1)th iteration. and minimum main lobe level The specific process is as follows:

[0110] definition The subproblem is transformed into:

[0111]

[0112] First fix The following formula can be used to solve for it. :

[0113]

[0114] The solution Substituting the transformed subproblem, we obtain a problem that only relates to the variable. Related sub-problems:

[0115]

[0116] Among them, when hour, ,otherwise ;when hour, ,otherwise Define separately and for and First, obtain an ascending set and remove duplicates; then merge the two sets, sort them in ascending order, and remove duplicates to obtain... ,make , , Introduction Only with variables The relevant subproblems are transformed into a piecewise function:

[0117]

[0118] in .make , , ,but

[0119]

[0120] The first derivative of this function is

[0121]

[0122] make , can be obtained

[0123]

[0124] The two roots of the equation are respectively

[0125]

[0126] In the interval The minimum value can be obtained by the following formula.

[0127]

[0128] The optimal minimum main lobe level obtained by selecting the global minimum value from all R+1 local piecewise functions is:

[0129]

[0130] Step 5.5: Obtain the dual variable for the (k+1)th iteration using the following formula. , and penalty factor and :

[0131]

[0132] Step 5.6: Determine whether the iteration termination condition is met, i.e., whether it is satisfied. and , This is the termination tolerance of the algorithm. If it is satisfied, the loop stops, and the simulated weights obtained in the current iteration are used. and numerical weights As the optimal simulated weights and digital weights; otherwise, proceed to step 5.1.

[0133] Example

[0134] The method for precise control of the emission pattern of a simulated digital hybrid array is further illustrated by Matlab simulation.

[0135] 1) Simulation system parameter settings

[0136] Unless otherwise specified, the HAD array used in each simulation is... , All array elements are evenly distributed and the spacing between array elements is [missing information]. Spatial angular domain Divided into 181 intervals Discrete grid, and set , , , , , and The array's mixed weight variables are randomly initialized. In addition, the FD array and FA array are selected as references for the HAD array.

[0137] 2) Beam plotting

[0138] To visually demonstrate the transmit beamforming effect of the HAD array, this embodiment uses the method proposed in this invention to plot the transmit beam pattern, and compares it with the typical HAD transmit pattern synthesis method: the two-stage method. The horizontal axis of the beam pattern represents the angular range of [-90°, 90°], and the unit of the vertical axis of the beam pattern is dB.

[0139] 3) Measurement indicators

[0140] In this invention, it is necessary to measure the effect of the final beamforming. The peak sidelobe level (PSL) and minimum mainlobe level (MML) are used to quantitatively measure the sidelobe and mainlobe levels of the transmitted beamforming. The definitions of PSL and MML are as follows:

[0141]

[0142]

[0143] in: This is the side lobe region. Main lobe region This is the level formula. Under the same initial conditions, the smaller the PSL, the better the performance of the radar and communication system in suppressing sidelobes during transmitted beamforming; the larger the MML, the better the performance of the radar and communication system in detecting the main lobe during transmitted beamforming.

[0144] 4) Results Analysis

[0145] Two simulation examples were conducted for this invention: a single main lobe flat-top beam and a double main lobe flat-top beam. Figure 2 It is a single main lobe flat-top beam. Figure 3 It is a double main lobe flat-top beam.

[0146] pass Figure 2 As can be seen, both the two-stage method and the proposed method can effectively form the main lobe and side lobes. The MML of the proposed method reaches 26.36 dB, the same as that of the two-stage method, but the main lobe ripple of the two-stage method is much larger than that of the proposed method. On the other hand, the MMLs of the FA and FD arrays are 24.39 dB and 26.38 dB, respectively. Furthermore, the PSL of the proposed method is -0.37 dB, only 1.77 dB higher than that of the FD array. Therefore, the HAD array achieves a good trade-off between beam-matching performance and hardware complexity.

[0147] pass Figure 3As can be seen, under the HAD array, the two-stage method fails to form an effective beam, while the method proposed in this invention can effectively form a beam and produces lower PSL and higher MML compared to the two-stage method. Furthermore, compared to the FD array, the method proposed in this invention shows almost no gain loss in the main lobe region, with only a 1.98 dB higher PSL. In radar applications, this loss is acceptable because main lobe gain is one of the most important indicators for long-range target detection.

[0148] In summary, the method described in this invention exhibits excellent overall performance. Compared to two-stage methods, the proposed method can effectively form a beam, precisely control the main lobe, and ensure a low PSL (Power Segmentation Level). When applied to radar and communication systems, this invention can significantly reduce hardware costs and computational complexity with minimal performance loss, demonstrating high practical value.

Claims

1. A method for precise control of the transmission pattern of a hybrid analog-digital array, characterized in that, Includes the following steps: Step 1: For the precise control of the transmit pattern of an HAD array, taking the maximization of the ratio of the minimum main lobe level to the peak sidelobe level as the criterion, the non-convex optimization problem model for precise control of the transmit pattern is as follows: ， in, The discrete angles represent the side lobe regions. This is the side lobe region. Discrete angles of the main lobe region Main lobe region For angle HAD array steering vector at the location, The HAD array is a mixed weight vector. For the digital weights of the HAD array, For the simulated weights of the HAD array, The number of digital subarrays. The number of elements in each simulated subarray. express A dimensional vector of all 1s; Step 2: Control the main lobe ripple within the set range, and introduce auxiliary variables to further express the non-convex optimization problem model of Step 1 as follows: ， in, Indicates the minimum main lobe level. Indicates the peak sidelobe level. Indicates the main lobe ripple. and These represent the number of discrete angle points in the main lobe region and the side lobe region, respectively. Step 3: Introduce auxiliary variables and ,definition , Constructing the augmented Lagrange function ， in, , and These are introduced auxiliary variables. and It is a dual variable. and Let the penalty factor be ; then the optimization problem model after step 2 is rewritten as: ， Step 4: Based on the Alternating Direction Penalty Method (ADPM) framework, the optimization problem model in Step 3 is decomposed into several easily solvable subproblems, specifically: ， ， ， ， ， ， ， In this context, all variables with a superscript of k+1 represent the results of the k+1th iteration, and all variables with a superscript of k represent the results of the kth iteration. , , and Let be the penalty factor transformation coefficient, and satisfy . , , This represents the maximum value of the penalty factor; Step 5: Solve the subproblems in Step 4, including simulating the weights. The solution is obtained iteratively using the backtracking projection gradient method, with digital weights. The closed-form solution is given using the Lagrange multiplier method, with auxiliary variables... and Closed-form solutions are given for all cases; finally, the optimal digital weights and analog weights are obtained to achieve the problem of precise control of the emission pattern of the HAD array.

2. The method for precise control of the transmission pattern of a hybrid analog-digital array according to claim 1, characterized in that: In step 1, the far-field narrowband signal of a HAD uniform linear array with a partially connected structure is considered, wherein each analog subarray is composed of... It consists of several isotropic antenna elements, with the spacing between each element being [missing information]. Each analog subarray is connected to one RF link; the HAD array has a total of M digital RF links. The steering vector of the HAD array is represented as ， in, Indicates the analog subarray at the angle The guide vector at that location, The steering vector representing the digital link. Indicates the carrier wavelength; Define the weight of the nth element on the mth simulated subarray as: , , The simulated weights of the HAD array are then expressed as: Define the RF link weight corresponding to the m-th analog subarray as: The digital weights of the HAD array are then expressed as: The mixed weights of the HAD array are equivalently represented as follows: ， The radiation pattern of the HAD array is then represented as follows: ， The simulated weights in the HAD array must satisfy the following constraints: ， The digital weights of the HAD array must satisfy the total power constraint: 。 3. The method for precise control of the transmission pattern of a hybrid analog-digital array according to claim 1, characterized in that, The problem transformation process in step 2 is as follows: A main lobe ripple constraint is applied to the main lobe angle region of the transmission pattern, as follows: ， in, This indicates ripples in the main lobe region. The total number of discrete points representing the angle of the main lobe region; considering the main lobe ripple constraint, the optimization problem in step 1 is further expressed as: ， Introducing auxiliary variables This represents the minimum gain in the main lobe region. If we represent the peak sidelobe level, then the above problem transforms into: 。 4. The method for precise control of the transmission pattern of a hybrid analog-digital array according to claim 1, characterized in that, The specific transformation process of the optimization problem after the equivalence in step 3 is as follows: For the non-convex optimization problem in step 2, the alternating direction penalty method is used for solution. The specific process is as follows: First, auxiliary variables are introduced. and ,make , The non-convex optimization problem in step 2 can then be rewritten as: ， Then, the sidelobe level constraints and main lobe ripple constraints are added as penalty terms to the objective function, resulting in the following Lagrangian function for the above problem: in , and These are introduced auxiliary variables. and It is a dual variable. and The penalty factor is used; finally, the optimization problem model after step 2 is rewritten as: 。 5. The method for precise control of the emission pattern of a hybrid analog-digital array according to claim 1, characterized in that, The specific process for solving each sub-problem in step 5 is as follows: Step 5.1: Ignore the constant term and... Equivalent representation is In the formula, all variables with superscript k represent the k-th iteration, and , ， ； The transformed subproblem is solved using the backtracking projection gradient descent method to obtain the simulated weights for the (k+1)th iteration. Step 5.2: Solve Obtain the numerical weight for the (k+1)th iteration; Step 5.3: Solve Obtain the auxiliary variables for the (k+1)th iteration. and peak sidelobe level ; Step 5.4: Solve Obtain the auxiliary variables for the (k+1)th iteration. and minimum main lobe level ; Step 5.5: Obtain the dual variable for the (k+1)th iteration using the following formula. , and penalty factor and : ， Step 5.6: Determine whether the iteration termination condition is met, i.e., whether it is satisfied. and , This is the termination tolerance of the algorithm. If it is satisfied, the loop stops, and the simulated weights obtained in the current iteration are used. and numerical weights Use the best simulated weights and digital weights; otherwise, return to step 5.

1.

6. The method for precise control of the transmission pattern of a hybrid analog-digital array according to claim 5, characterized in that, The specific process of solving the transformed subproblem using the backtracking projection gradient descent method to obtain the simulated weights for the (k+1)th iteration is as follows: Step 5.1.1: Obtain the objective function for the j-th iteration. The gradient is Its negative gradient direction is ; Step 5.1.2: Adaptively update the step size using backtracking search. The specific method is as follows: Initialize step size Using the formula Iteration step size, This is the step size coefficient. When the backtracking condition is met, the iteration stops, and the current iteration position is set to... The final step size for the j-th round is defined by the backtracking condition as follows: ， Indicates the vector Each element is projected onto the unit circle, i.e. ; Step 5.1.3: Solve for the simulated weights in the (j+1)th iteration using the following formula: ， Determine whether the iteration termination condition is met, i.e., whether it is satisfied. , This is the algorithm's termination tolerance; if it is met, the loop stops, and the current value is set to zero. Use the optimal simulation weight for the (k+1)th round; otherwise, return to step 5.1.

1.

7. The method for precise control of the transmission pattern of a hybrid analog-digital array according to claim 5, characterized in that: In step 5.2, the specific process of solving the transformed digital weight subproblem using a closed-form solution to obtain the digital weights for the (k+1)th iteration is as follows: Ignoring the constant term, the subproblem can be equivalently represented as ， in, , , Represents an N x 1 column vector. Indicates the vector Diagonalize to a matrix; define the Lagrange function as: ， in, They are Lagrange multipliers; according to the first-order optimality condition of the Lagrange function, we can obtain... ， definition The matrix decomposition is as follows ,in, A matrix composed of eigenvectors. Let represent a diagonal matrix composed of eigenvalues, and Redefining and order Then the equation simplifies to ， Solving for the optimal Lagrange multipliers using the line search method Substitute the Lagrange multipliers into the following formula: ， Obtain the numerical weights for the (k+1)th iteration. .

8. The method for precise control of the transmission pattern of a hybrid analog-digital array according to claim 5, characterized in that, The following subproblems are solved using closed-form solutions. Obtain the auxiliary variables for the (k+1)th iteration. and peak sidelobe level The specific method is as follows: Ignore irrelevant terms and let The subproblem is transformed into: ， First fix The following formula can be used to solve for it. : ， when hour, ,otherwise ; Define respectively for Given an ascending set and removing duplicates; let , , Introduction , The objective function in the subproblem is transformed into a piecewise function: ， in Represented as: ， The first derivative is ， make , , ,but The two roots are ， Then piecewise function In the interval The minimum value is obtained by the following formula. The optimal peak sidelobe level for the (k+1)th iteration is selected from all L+1 local piecewise functions as the global minimum. ， Will Substitute and solve In the formula, we obtain the auxiliary variable for the (k+1)th iteration. .

9. The method for precise control of the emission pattern of a hybrid analog-digital array according to claim 5, characterized in that: The following subproblems are solved using closed-form solutions. Obtain the auxiliary variables for the (k+1)th iteration. and minimum main lobe level The specific method is as follows: definition The subproblem is transformed into: ， First fix The following formula can be used to solve for it. : ， The solution Substituting the transformed subproblem, we obtain a problem that only relates to the variable. Related sub-problems: Among them, when hour, ,otherwise ;when hour, ,otherwise ; Define respectively and for and First, obtain an ascending set and remove duplicates; then merge the two sets, sort them in ascending order, and remove duplicates to obtain... ,make , , Introduction Only with variables The relevant subproblems are transformed into a piecewise function: ， in ;make , , ,but ， The first derivative of this function is ， make The two roots of the equation are: ， In the interval The minimum value is obtained by the following formula. ， The optimal minimum main lobe level for the (k+1)th iteration is selected from all R+1 local piecewise functions as the global minimum. 。

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